Number Theory Seminar: Dragos Ghioca

Abstract:

In early 1960's, Lang proved that if for a given polynomial G(X,Y) with complex coefficients, there exist infinitely many pairs (x,y) where both x and y are roots of unity such that G(x,y) = 0, then essentially G(X,Y) = X^mY^n - c, for some integers m and n, and a root of unity c. In 2009, Masser and Zannier proved a result (similar in the spirit of Lang's result) for torsion points on a family of elliptic curves. In our talk we explain how both results come from the same general principle in arithmetic geometry, and at the same time we present a partial result to a more general conjecturewhich subsumes both Lang and Masser-Zannier theorems.